N78-14571

A SOLUTION TO THE PROBLEM OF SAR RANGE CURVATURE

R. K. Raney

Canada Centre for Remote SensingOttawa, Ontario

ABSTRACT

When synthetic aperture radar systems are pushed to attain finer resolutionat larger ranges than was previously the case for remote sensing purposes, thegeometric signal aberration known as range curvature arises. Known techniquesfor correcting range curvature are exact at only one selected range, thusforcing neighboring ranges to use the same correction as an approximation. (ForSeasat A, four separate correction filters would be required over the designswath width.) This paper outlines a solution to the problem that is exact atall ranges, thus simplifying and improving the image processing for such systems.

1. INTRODUCTION

Synthetic aperture radar systems achieve beam sharpened azimuth resolutionby coherently exploiting the incremental changes in range between the sensorand an object as it moves through the beam, as shown in Figure 1. Range curva-ture arises if the differential range <5R that is beneficial becomes larger than(half of)the range resolution pr, at which time the resulting signal accumulatesan aberration that may prevent the desired range resolution from being achieved.

The radar/object range as a function of time is given by2

R(t) = Ro + % (Vt) (1)RO

using the first two terms of the Taylor series expansion. The second term ofEq_n. 1 (the differential range) takes on its maximum value for the largest valueot" aspect angle 9 which we let be 9ft.

Max 6R = % RO (TM (2)

which is linearly increasing with range. Likewise, the achievable azimuthresolution pa is proportional to 9^, so we see that range curvature is com-pounded by greater ranges on finer resolution.*

*In the broadside case, 9^= 7- - so that range curvature,which arises if ,._ ^

Max <5R

imposes the SAR constraintthat if RO igD2 -

> Dr then range curvature shouldbe corrected.

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Previous techniques used to correct for range curvature are given anexcellent summary in Leith (1) . These techniques have in common a recognitionof the range dependence of curvature described by Eqn. 1. However, it turnsout that for optical implementations it is more convenient to work either withthe range Fourier transform of (the ensemble of the) functions represented byEqn. 1, or with the two dimensional transform. In these approaches, the(transformed) range data are dispersed and superimposed, thus one has no choicebut to choose a correction that is exact at only one range, and to accept theconsequential approximation that results for adjacent ranges. Implementationmay be elegant, but in effect a compromise.

2. THE SOLLUTION

The key to the solution of the range curvature problem rests in recognitionof the coordinate system in which the aberration arises within the sensor/objectspace itself. From Figure 1, the incremental curvature is a single valuedfunction determined by (i) the (instantaneous slant) range to an object, and(ii) its position (with respect to same reference angle) within the real antennabeam in the so-called azimuth coordinate. Operationally, a convenient azimuthposition reference is the zero-Doppler line,* and in the simple case modelledhere, there is a one-to-one equivalence between angular bearing and Dopplerfrequency for given object within the antenna pattern at a particular point intime. Indeed, Doppler frequency fj is given by

= -|- - r- R(t) (3)

so that from Eqn . 1 we have

<•<» ' - wwhich relates for each and every object the time history and its Doppler historyas it traverses the beam.

The signal history of relevance to the correction problem is to be foundin the range image/azimuth Doppler plane. Thus, on reception of SAR data, werequire two transformations: range focusing and azimuth frequency (Fourier)transformation. The loci of signals in the (r,f(j), plane is given by Eqn. 1,with a linear change in variables according to Eqn. 4, yeilding

R(fd) - Ro [l + %(z )2 fd]

as plotted in Figure 2.

This apparently trivial step has many ramifications. First, it is con-ditional on there being sufficient signal structure (large time-bandwidthproduct) so that the one-one mapping from time to frequency is valid. This isthe case for Seasat A, and other SAR devices in the class we are considering.Second, the correction to be accomplished is explicitly represented by the secondterm of Eqn. 5, which is range and Doppler dependent. Since it is gentlyquadratic in Doppler and linear in range, it may be realized by optical ordigital devices, and is exact at all ranges so compensated. Third, the range/Doppler plane(surprisingly!) does not occur in conventional anamorphic telescopicoptical processors normally used for SAR, although there is no fundamentalreason forbidding it! Thus, this implementation would require modification ofthe current methods, although substantially the same optical elements could beemployed. Fourth, in the event several sub-images are to be integrated, each

*Note that for the Seasat case the "zero-Doppler line is neither perpendicularto the satellite velocity vector, nor is it a straight line in the slant rangeplane.

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from a different aspect angle within the real antenna beam, implementation ofthe (r, fd) range curvature correction as suggested herein automatically bringsabout the required registration of these multi-look images, thereby solvinganother problem facing these systems.

3. CONCLUSIONS

Use of the range image/azimuth Doppler plane allows full correction of theSAR range curvature problem, correct at all ranges and in such a way that multi-look images derived from sub-apertures become automatically registered. Theapproach is realizable either optically or digitally, although analysis showsthat for the Seasat case, in which the model is more complex than that shown here,digital methods are more flexible.

The author would like to acknowledge his indebtedness to collegues from theCanada Centre for Remote Sensing, the Communications Research Centre, andMacDonald, Dettweiler and Associates with whom many probing discussions havetaken place on this and related topics.

REFERENCE

(1) E . N . Leith JOSA 63 No .2 , Feb. 1973;pp 119-126.Apparent translation of object

past sensor >.

Velocity vector ofsensor platform

incremental range change 6R

Figure 1. Geometry. Coherent PulseRanging System in the sensor/obiecttrajectory plane. t=o at the abeamor 0=o position.

Currature a firiction ofraige, flight parameters.

Doppler bounds are a function of antennageometry, sensor and scene motion \ectors

Figure 2. Loci of scatterer range plotted as a function of Douplerfrequency shift, for a variety of reference ranges Ro.

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